1. Field of the Invention
The present invention relates to a method and an apparatus for calculating the transfer characteristic of an imaging optical system and an exposure apparatus using the calculated transfer characteristic.
2. Description of the Related Art
Assume that an imaging optical system is linear and shift-invariant, as shown in FIG. 15. Let f(x) be the input signal to the optical system, h(x) be the transfer characteristic of the optical system, and g(x) be the output signal after imaging in the optical system. Then, g(x) is generally given by:
                                                                        g                ⁡                                  (                  x                  )                                            =                                                f                  ⁡                                      (                    x                    )                                                  *                                  h                  ⁡                                      (                    x                    )                                                                                                                          =                                                ∫                                      -                    ∞                                    ∞                                ⁢                                                                            h                      ⁡                                              (                        τ                        )                                                              ·                                          f                      ⁡                                              (                                                  x                          -                          τ                                                )                                                                              ⁢                                      ⅆ                    τ                                                                                                          (        1        )            
As indicated by equation (1), g(x) is expressed by a convolution of f(x) and h(x). If the transfer characteristic is known, so-called image reconstruction is prevalently done, in which the input signal is reconstructed from the output signal by using an inverse filter or Wiener filter. To obtain the transfer characteristic, a method of directly obtaining an impulse response by inputting an impulse signal (delta function) to the system as an input signal is generally used.
Along with recent performance improvement and cost reduction of electronic devices, a semiconductor exposure apparatus is required to accurately and to efficiently manufacture semiconductor devices to be incorporated in the electronic devices. An exposure apparatus for forming a semiconductor circuit pattern by exposure is also required to perform accurate and efficient manufacturing. An exposure apparatus for producing semiconductor devices transfers a circuit pattern formed on a reticle or mask (to be referred to as a “reticle” hereinafter) to a wafer or glass plate (to be referred to as a “wafer” hereinafter) with a photosensitive material applied. Generally, to accurately form a circuit pattern by exposure, it is important to accurately align a reticle relative to a wafer.
In a conventional alignment method, alignment marks are exposed and transferred onto a wafer simultaneously with exposure/transfer of a circuit pattern formed on a reticle. An alignment detection optical system detects the positions of a plurality of preset alignment marks from all shots of the alignment marks so that position measurement is sequentially performed. The position measurement result is statistically processed to calculate the arrangement of all shots. Based on the calculation result, the wafer is aligned relative to the reticle.
The alignment marks serve as an index to accurately align the reticle and wafer. The alignment marks are also required to be precise along with size reduction of circuit patterns. In recent years, semiconductor manufacturing technologies such as CMP (Chemical Mechanical Polishing) have been introduced. Accordingly, errors (WIS: Wafer Induced Shift) occur due to the wafer process, including a variation in alignment mark shape between wafers or shots, resulting in degradation in alignment accuracy.
Japanese Patent Laid-Open No. 2004-117030 discloses a technique of correcting WIS by offset correction. “Offset correction” is a method of calculating an offset amount, i.e., a shift amount between the proper position (true value) of an alignment mark and an alignment mark position actually detected by a detection system and executing correction based on the offset amount.
Japanese Patent Laid-Open Nos. 6-151274 and 8-94315 disclose techniques related to a position detection method.
However, such an offset amount is generated by factors other than the error (WIS) caused by the wafer process as well. For example, an error (TIS: Tool Induced Shift) caused by the exposure apparatus (alignment optical system) or an error (TIS-WIS Interaction) caused by interaction between TIS and WIS may also degrade the alignment accuracy. WIS includes an alignment mark step offset, asymmetry, and resist application variation. TIS includes coma or spherical aberration of an alignment optical system.
The NA of a recent alignment optical system is high, though it cannot completely eliminate TIS. For this reason, if WIS is present due to the TIS-WIS interaction (e.g., a low step mark), the offset amount increases and makes it impossible to accurately detect the position of an alignment mark. Referring to FIGS. 5A and 8B, even in the same optical system, the offset amount in an alignment mark with a low step, as shown in FIG. 8B is larger than that in an alignment mark with a normal step, as shown in FIG. 8A because of the existence of TIS.
The above-described transfer characteristic h(x) includes the error (TIS) caused by the apparatus. If it is possible to calculate the transfer characteristic and to reconstruct the input signal from the output signal by using an inverse filter or Wiener filter, the influence of TIS in the reconstructed input signal is minimal. Hence, the offset amount by the TIS-WIS interaction is expected to be small.
The challenge is therefore to accurately calculate the transfer characteristic of the optical system. In the general method of directly measuring an impulse, however, it is necessary to form the intensity distribution of a δ function at the observation position. It is actually difficult to form the distribution (impulse width) of a δ function. Hence, there is a limit to the distribution width that can be formed, and an error is generated. The δ function preferably has a distribution which is as narrow as possible. However, if the distribution is too narrow, the energy becomes low, resulting in a poor S/N ratio. This also generates errors.